Violation of a Bell inequality in two-dimensional state spaces for radial quantum number

The Bell inequalities for azimuthal quantum number, i.e., orbital angular momentum, have been extensively reported to show quantum correlations between the spatial degree of freedom of entangled photon pairs. Here we investigate instead the Bell inequalities for radial quantum number. We demonstrate, both theoretically and experimentally, the violation of a suitable Clauser-Horne-Shimony-Holt Bell inequality for the radial index p of the Laguerre-Gaussian mode, thus manifesting the entanglement in the radial structure of two-photon wave functions.